A111340 Number of positive integer 2-friezes with n-1 nontrivial rows.

1, 5, 51, 868, 26952

Offset: 1

Author

N. J. A. Sloane, based on correspondence from James Propp, May 08 2005

Keywords

• nonn
• fini
• full

Comments

The n-th term is the number of positive integer tables a(i,m) (with i running from 1 to n+3 and m running from minus infinity to infinity) subject to the boundary conditions a(i,m) = 0 when i = 1 or i = n+3 and a(i,m) = 1 when i = 2 or i = n+2 and the internal condition a(i,m-1) a(i,m+1) = a(i-1,m) a(i+1,m) + a(i,m) when i is strictly between 2 and n+2.

It is not known as of this writing whether any or all of the terms of the sequence beyond 868 are finite. If the final term "a(i,n)" in the internal condition is replaced by "1", then what we are looking is just a frieze pattern a la Conway and Coxeter (or rather two interlaced frieze patterns that do not interact at all).

According to the lecture notes by S. Morier-Genoud (see paragraph "2-frieze of positive integers"), a(5) is conjectured to be 26952, and it is proved that there are no more finite terms. - Andrei Zabolotskii, Nov 01 2022

Examples

			The number 1 in the sequence is counting the rather boring configuration
    0 0 0 0 0 0 0 0
... 1 1 1 1 1 1 1 1 ...
    1 1 1 1 1 1 1 1
    0 0 0 0 0 0 0 0
The number 5 is counting the configuration
    0 0 0 0 0 0 0 0 0 0
    1 1 1 1 1 1 1 1 1 1
... 1 1 2 3 2 1 1 2 3 2 ...
    1 1 1 1 1 1 1 1 1 1
    0 0 0 0 0 0 0 0 0 0
and its four distinct cyclic shifts, each of which repeats with period 5 (note the Lyness 5-cycle A076839 in the middle).
a(2) = A000108(3) = number of friezes of type A_2 (cyclic shifts of A139434), a(3) = A247415(4). a(4) and a(5) also count friezes of types resp. E_6 and E_8.
		

Links

• Sophie Morier-Genoud, Lecture notes on Integrable Systems and Friezes, 2017.
• Robin Zhang, A positive Siegel theorem: Dynkin friezes and positive Mordell-Schinzel, arXiv:2503.08800 [math.NT], 2025.

Crossrefs

Cf. A000108, A247415, A076839, A139434.

Extensions

The last finite term, a(5), added based on Zhang's preprint and name clarified by Andrei Zabolotskii, May 14 2025